Solve the exponential equation for $x$. 2 4 x − 5 ⋅ 32 x + 1 = 2 6 x − 7 2\^{ 4x-5}\cdot 32\^{ x+1}=2\^{ 6x-7} $x=$
Explanation: The strategy Let's write $32$ in base $2$. Then, using the properties of exponents, we can express the entire left hand side of the equation as $2$ raised to some linear function. Finally, we can equate the exponents of the resulting equation to solve for the unknown. Simplifying the left hand side 2 4 x − 5 ⋅ 32 x + 1 = 2 4 x − 5 ⋅ ( 2 5 ) x + 1 = 2 4 x − 5 ⋅ 2 5 x + 5 = 2 4 x − 5 + ( 5 x + 5 ) = 2 9 x ( 32 = 2 5 ) ( ( a n ) m = a n ⋅ m ) ( a n ⋅ a m = a n + m ) \begin{aligned} 2\^{ 4x-5}\cdot 32\^{ x+1}&=2\^{ 4x-5}\cdot (2^5)\^{ x+1}&&&&(32=2^5)\\\\ &=2\^{C{4x-5}}\cdot 2\^{ {5x+5}}&&&&((a^n)^m=a^{n\cdot m}) \\\\ &=2\^{ C{4x-5} \ + \ ({5x+5}) }&&&&(a^n\cdot a^m=a^{n + \normalsize m})\\\\ &=2\^{ 9x} \end{aligned} Solving the linear equation We obtain the following equation. 2 9 x = 2 6 x − 7 2\^{ 9x}=2\^{ 6x-7} Now we can equate the exponents and solve for $x$. $\begin{aligned} 9x &=6x-7\\\\ x &= -\dfrac{7}{3}\end{aligned}$ The answer The answer is $x=-\dfrac{7}{3}$. You can check this answer by substituting $\it{x=-\dfrac{7}{3}}$ in the original equation and evaluating both sides.